On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

Chen, Jinsong ; Tan, Yijia

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1117-1129.

Résumé

Let $n$ be a positive integer, and $C_{n} (r)$ the set of all $n\times n$ $r$-circulant matrices over the Boolean algebra $B=\lbrace 0,1\rbrace $, $G_{n}=\bigcup _{r=0}^{n-1}C_{n}(r)$. For any fixed $r$-circulant matrix $C$ ($C\ne 0$) in $G_{n}$, we define an operation “$\ast $” in $G_{n}$ as follows: $A\ast B=ACB$ for any $A,B$ in $G_{n}$, where $ACB$ is the usual product of Boolean matrices. Then $(G_{n},\ast )$ is a semigroup. We denote this semigroup by $G_{n}(C)$ and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix $C$. Let $F$ be an idempotent element in $G_{n}(C)$ and $M(F)$ the maximal subgroup in $G_{n}(C)$ containing the idempotent element $F$. In this paper, the elements in $M(F)$ are characterized and an algorithm to determine all the elements in $M(F)$ is given.

Détail

Classification : 06F30, 15A33, 15A36
Keywords: generalized ciculant Boolean matrix; sandwich semigroup; idempotent element; maximal subgroup